0: Introduction

This is a set of notes I developed for an e-learning course in Algebraic Structures offered by Maseno, University in Western Kenya. The idea is to introduce the key concepts of algebraic structures without assuming much background in higher mathematics. Math education in Kenya is heavy on calculation (it's relatively easier to teach and evaluate), but often falls short when it comes to teaching students to think creatively about mathematics, and really understand the subject as it relates to the world beyond the test. On the bright side, these same students are usually very ready to take a more creative approach to mathematics: good skills in calculation provides at least a good intuition for working with numbers, and gives a good foundation from which to build. Kenyan students are also generally very enthusiastic when presented with interesting mathematics.

The notes are trying, then, to accomplish the following:

Give students a first encounter with algebraic structures: Groups, rings, fields, and vector spaces,

Create an intuition for how these objects appear 'in the world,' meaning both in the real world and in the broader scope of mathematics,

Encourage students to engage with the material in a creative way, and

Teach/Reinforce important points from the foundations of mathematics, such as induction.

It's a lot to ask for a single ten-week term. Let's see where we get.

The notes themselves are divided into eleven 'chapters,' one for each week of Maseno's term, plus this introductory chapter. Taking a cue from computer science, all numbering of chapters and sections starts at 0. As the course becomes fully developed, I will be inserting videos for each section, giving an alternate presentation of the ideas. But the text is primary!

Here are some underlying principles that I believe strongly in, which also guide the formation of these notes.

We live in the future. Computers are somewhere between a million and a billion times faster at computation than humans are. Therefore, we should focus our teaching on what humans do better than computers: Understanding, problem solving, and placing things in context. It is often essential to understand how to compute things (indeed, otherwise we would not be able to tell the computer how to do computations for us!), but computation should not be the aim of a course.

We live in the future. We can communicate at almost zero-cost at slightly less than the speed of light. Information is governed by post-scarcity economics, and we need to treat it as such. This means we cannot treat information like a scarce resource to be hoarded: we must share our infinite wealth freely. Thus, these notes will remain free, and will be distributed under the Gnu Public License.

Design Principles

This book is also a programming project! As of this writing, I'm learning some modern web-programming tools; this book runs on Django, HTML5, Javascript, JQuery, MathJAX, the Sage Cell Server, and probably more by the time I'm done. HTML5 support is becoming more common in browsers, and should be an available standard for a long time to come.

Here is a list of design principles that I hope to adhere to for the final product:

An important principle of the book is to support multiple learning modes: there should be a combination of video and text for every section.

Videos, for their part, should be no longer than five or ten minutes long. Likewise, sections of the text should be somewhere south of 1000 words.

Each section should have at least one exercise, and these exercises should encourage both basic mechanical understanding and encourage creative approaches to the material.

The finished work should be free and freely available.

The finished work should meet the standards of a Maseno University e-learning course; in particular, have at least ten 'topics' to be digested at a rate of one-per-week, with clearly marked exercises to act as assignments.

Wherever reasonable, interactive elements (probably using Sage) should be included.